Finding Large Sticks and Potatoes in Polygons.
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Transcript of Finding Large Sticks and Potatoes in Polygons.
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Finding Large Sticks and Potatoes in Polygons.
Olaf Hall-HoltSt. Olaf College
Matya Katz and Arik Sityon Ben-Gurion University
Joseph S.B. MitchellStony Brook University
Piyush KumarFlorida State University
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1. Natural Optimization Problems
2. Shape Approximation
3. Visibility Culling for Computer Graphics
Motivation
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Biggest Potato
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Peeling Potato inside Smooth Closed Curves
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Biggest French Fry
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Longest Stick
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Related and Prior Work
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Convex Polygons on Point Sets
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Related Work: Longest Stick
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Our Results (On Peeling)
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1. Divide and Conquer Algorithm
2. Uses balanced cuts (Chazelle Cuts)
Approximate Largest Stick
e
a
b c
db
cd
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Approximate Largest Stick
1. Compute weak visibility region from anchor edge
(diagonal) e.
2. (p) has combinatorial type (u,v)
3. Optimize for each of the O(n) elementary intervals.
Theorem:
One can compute a ½-approximation for longest stick in a simple polygon in O(nlogn) time.
Algorithm:
At each level of the recursive decomposition of P, compute longest anchored sticks from each diagonal cut: O(n) per level.
Longest Anchored stick is at least ½ the length of the longest stick.
Open Problem:Can we get O(1)-approx in O(n) time?
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Approximate Largest Stick: Improved Approx.
Algorithm:
Bootstrap from the O(1)-approx, discretize search space more finely, reduce to a visibility problem, and apply efficient data structures
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Pixels and the visibility problem.
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Pixels and the visibility problem.
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Approximate Largest Stick
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Approximate Largest Convex-gon
1. Suffices to look for a large triangle to get a O(1)-
approximation.
2. For any convex body B, there is an inscribed triangle T*
of area at least c.area(B). There exists a O(1)
approximation to T* anchored at a cut computable in
O(nlogn).
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Approximate Largest Convex-gon
1. Suffices to look for a large triangle to get a O(1)-
approximation.
2. For any convex body B, there is an inscribed triangle T*
of area at least c.area(B). There exists a O(1)
approximation to T* anchored at a cut computable in
O(nlogn).
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Approximate Largest Triangular potato
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Approximate FAT Largest triangular potato
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Approximate Fat Triangles : Results
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A Sampling approach
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Largest Area Triangle using Sampling
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Largest Area Triangle by Sampling: A difficulty
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Peeling an ellipse
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Max Area ellipse inside sampled curves
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Linearized convex hull + Normal cond. + Inside Test
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An Example output
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An Example output
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• PTAS for largest triangle ?
• Find exact solutions/approximations for biggest potato ?
• Packing convex sets in shapes.
• Sub quadratic bounds for max area star shaped
polygons?
• Find k convex potatoes to max the area of the union?
Sum? Max area k-gon (Non-convex)?
• d-D?
Future WorkQuestions?